Related papers: Efficient and Stable Multi-Dimensional Kolmogorov-…
Suppose we have a sample from a distribution $D$ and we want to test whether $D = D^*$ for a fixed distribution $D^*$. Specifically, we want to reject with constant probability, if the distance of $D$ from $D^*$ is $\geq \varepsilon$ in a…
Statistical testing is widespread and critical for a variety of scientific disciplines. The advent of machine learning and the increase of computing power has increased the interest in the analysis and statistical testing of…
We present an algorithm that takes a discrete random variable $X$ and a number $m$ and computes a random variable whose support (set of possible outcomes) is of size at most $m$ and whose Kolmogorov distance from $X$ is minimal. In addition…
We give an estimate for the Kolmogorov distance between an infinitely divisible distribution (with mean zero and variance one) and the standard Gaussian distribution in terms of the difference between the fourth moment and 3. In a similar…
Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identically distributed (i.i.d.) draws from a user-specified probability distribution known…
One of the major problems in Machine Learning (ML) and Artificial Intelligence (AI) is the fact that the probability distribution of the test data in the real world could deviate substantially from the probability distribution of the…
We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein…
We present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball,…
Maximum Mean Discrepancy (MMD) has been widely used in the areas of machine learning and statistics to quantify the distance between two distributions in the $p$-dimensional Euclidean space. The asymptotic property of the sample MMD has…
We consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the…
We establish general upper bounds on the Kolmogorov distance between two probability distributions in terms of the distance between these distributions as measured with respect to the Wasserstein or smooth Wasserstein metrics. These bounds…
We present an efficient algorithm that, given a discrete random variable $X$ and a number $m$, computes a random variable whose support is of size at most $m$ and whose Kolmogorov distance from $X$ is minimal, also for the one-sided…
Statistical distances quantifies the difference between two statistical constructs. In this article, we describe reference values for a distance between samples derived from the Kolmogorov-Smirnov statistic $D_{F,F'}$. Each measure of the…
We survey some basic results on the Gromov-Prohorov distance between metric measure spaces. (We do not claim any new results.) We give several different definitions and show the equivalence of them. We also show that convergence in the…
We analyzed the effect of the deviation of the exact distribution of the p-values from the uniform distribution on the Kolmogorov-Smirnov (K-S) test that was implemented as the second-level randomness test. We derived an inequality that…
We extend the Kolmogorov--Smirnov (K-S) test to multiple dimensions by suggesting a $\mathbb{R}^n \rightarrow [0,1]$ mapping based on the probability content of the highest probability density region of the reference distribution under…
Integral probability metrics (IPMs) constitute a general class of nonparametric two-sample tests that are based on maximizing the mean difference between samples from one distribution $P$ versus another $Q$, over all choices of data…
In this contribution, we derive explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fr\'echet margins. We formulate those bounds in terms of (i) Wasserstein distances between de Haan representers, (ii)…
The two-sample Kolmogorov-Smirnov test is a widely used statistical test for detecting whether two samples are likely to come from the same distribution. Implementations typically recur on an article of Hodges from 1957. The advances in…
We consider the universality of the nearest neighbour eigenvalue spacing distribution in invariant random matrix ensembles. Focussing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution…